I painted the ears before I started on the dome and cheeks, but not for any real necessity - to be honest, I don't think there is any real science. I usually start with the back panels, as they are sort of an area by themselves, so once done you can cover them up, while the rest of the areas are all sorta connected in some way. But it really comes down to finding an order that works well for you, and makes the most sense to your own painting technique
With white carbon paper, personally - i'm not a huge fan. The white residue can be a little fiddly to remove, and I find standard charcoal coloured graphie paper works best for my needs. The print it leaves is graphite, and not as 'waxy' a feeling as the white. I have had very few issues with it, and even on dark colours, by turning it into the light, it is always visible. If any areas need to be removed, a very light steel wool or light rub with a decent eraser is usually all it takes..
If the print isn't very visible, that could actually be more the tools you are using to trace the templates. Standard pencils and soft leads will be the culprit here, so I always use a technical pencil, the kind you can insert refillable leads, and click the lead up from the bottom. Here in Australia, Pentel and Pacer brands make a bunch of them. I use a Pentel brand one that fits fine 0.5mm leads. I always go with the harder leads too, so generally a HB. The combo of a nice, consistent and sharp lead as well as it being fine and a hard lead, will provide you with clear and fine-lined stencils once transferred to your lid. It's only how I go about it all, but I seem to have few issues, so it seems a good option